Chemical Kinetics: Rate Laws, Order and Molecularity

Just as a car's speed is the distance covered divided by the time taken, a reaction's rate is the change in concentration of a reactant or product divided by the time taken for that change. Some reactions are essentially instantaneous, silver chloride precipitating the moment silver nitrate meets sodium chloride; others crawl along over years, iron slowly rusting in humid air; and many, like the inversion of cane sugar, proceed at a genuinely moderate, measurable pace. For a simple reaction R to P, the rate can be tracked either as the rate of disappearance of R, -delta[R]/delta t, or the rate of appearance of P, +delta[P]/delta t, and since reactant concentration is falling (delta[R] is naturally negative), a minus sign is included to keep the rate itself a positive number. Both expressions describe the exact same underlying process from two different vantage points, and both carry units of concentration divided by time, typically mol/(L.s).

Measuring butyl chloride's hydrolysis rate over successive time intervals reveals something important: the average rate isn't constant, it steadily falls from 1.90x10^-4 mol/(L.s) in the first 50 seconds down to 0.4x10^-4 mol/(L.s) between 700 and 800 seconds, since concentration itself keeps dropping. This means average rate can only describe a reaction's speed across some interval, never at one specific moment. To capture a genuine, single-instant speed, chemists shrink that time interval down toward zero, defining the instantaneous rate as the limit of average rate as delta t approaches zero, mathematically d[R]/dt, which is found graphically by plotting concentration against time and measuring the exact slope of a tangent line drawn at the moment of interest. Just as a car's speedometer reads its speed at this exact instant, not its average speed since the trip began, instantaneous rate is chemistry's equivalent, and it's genuinely the more useful, more fundamental quantity of the two.

For a reaction like Hg(l) + Cl2(g) to HgCl2(s), where every species has a coefficient of 1, the rate of disappearance of either reactant exactly equals the rate of appearance of the product, no adjustment needed. But for 2HI(g) to H2(g) + I2(g), HI disappears twice as fast as H2 or I2 appears, simply because two HI molecules are consumed for every one H2 and one I2 produced. To give this reaction one single, unambiguous rate, chemists divide each species' own rate of change by its own stoichiometric coefficient: Rate = -(1/2)(delta[HI]/delta t) = +delta[H2]/delta t = +delta[I2]/delta t. This same principle generalises to any reaction, however many species and however large the coefficients, ensuring that no matter which reactant or product a chemist happens to measure, they arrive at the exact same, single value for the reaction's overall rate.

The rate law (or rate expression) expresses reaction rate in terms of reactant concentrations, Rate = k[A]^x[B]^y, where k is the rate constant, and the crucial, easily-missed point is that the exponents x and y may or may not match the reaction's own stoichiometric coefficients a and b. For CHCl3 + Cl2 to CCl4 + HCl, experiment shows Rate = k[CHCl3][Cl2]^(1/2), an exponent of one-half on chlorine despite its coefficient being 1. For ethyl acetate's hydrolysis, Rate = k[CH3COOC2H5]^1[H2O]^0, water's exponent is zero despite it being a full reactant in the balanced equation. These mismatches aren't errors or exceptions; they're the norm, and they prove decisively that a rate law can never be predicted by inspection alone, no matter how carefully a chemist balances the equation. It must always, without exception, be determined experimentally.

Determining an actual rate law means running controlled experiments, changing one concentration at a time while holding others fixed, and watching how the initial rate responds. For 2NO(g) + O2(g) to 2NO2(g), doubling [NO] while holding [O2] fixed makes the rate jump by a factor of four, from 0.096 to 0.384 mol/(L.s), revealing a second-order dependence on NO, since doubling a squared term quadruples the result. Doubling [O2] while holding [NO] fixed simply doubles the rate instead, revealing a first-order dependence on O2. Combining both findings gives the full rate law, Rate = k[NO]^2[O2]^1, and in this particular case, the exponents do happen to match the balanced equation's own coefficients, 2 and 1, though as the previous example already showed, that match is coincidental, not guaranteed, and must still be confirmed by exactly this kind of systematic experiment.

The order of a reaction is simply the sum of every exponent appearing in its experimentally-determined rate law; for NO and O2's combined rate law, order = 2 + 1 = 3, a third-order reaction overall, while the individual values, 2 and 1, are each species' own order with respect to that specific reactant. Order can take genuinely any value: zero (rate completely independent of that reactant's concentration), a whole number like 1, 2, or 3, or even a fraction, as chlorine's one-half order already demonstrated. A zero-order reaction is a particularly interesting case, meaning the rate stays constant no matter how concentration changes, something that happens under special conditions, such as a reaction occurring on a catalyst's surface once that surface is fully saturated with reactant molecules, where adding more reactant simply can't speed things up any further, since the catalyst's own limited surface, not concentration, has become the real bottleneck.

The rate constant k's units aren't fixed; they change depending on the overall order of the reaction, since rate itself always carries units of concentration/time regardless of order, but the concentration terms multiplying k on the right side of the rate law change with order. Working through the algebra for a zero-order reaction gives k units of mol/(L.s); for a first-order reaction, the concentration terms cancel entirely, leaving k with units of simply s^-1 (time inverse); for a second-order reaction, k carries units of L/(mol.s), or equivalently mol^-1.L.s^-1. This relationship runs in reverse too, genuinely usefully: given only a rate constant's units, say s^-1 alone with no concentration dependence at all, a chemist can immediately identify that reaction as first order, without needing any further information about concentrations or mechanism.

Molecularity is the number of reacting species, atoms, ions, or molecules, that must collide simultaneously in a single elementary (one-step) reaction, and it can only ever be a whole number, 1 (unimolecular), 2 (bimolecular), or, very rarely, 3 (termolecular), since the odds of four or more species colliding at precisely the same instant are essentially negligible. Order, by contrast, is a purely experimental quantity, determined by measuring how rate responds to concentration, and it can be zero, fractional, or any whole number, with no upper limit imposed by physical collision probability. For a genuinely single-step elementary reaction, molecularity and order happen to coincide exactly, but for a complex, multi-step reaction, molecularity has no meaning applied to the overall reaction at all, only to each individual step within its mechanism, while order remains perfectly well-defined for the overall reaction throughout.

A reaction that appears, from its balanced equation alone, to require an implausible ten-molecule collision, like KClO3 + 6FeSO4 + 3H2SO4, is actually observed to be second order overall, immediate proof that it must proceed through multiple simpler steps rather than one impossible collision. Just as a relay team's overall time depends entirely on its slowest runner, a multi-step reaction's overall rate is controlled entirely by its slowest individual step, called the rate-determining step. Hydrogen peroxide's iodide-catalysed decomposition demonstrates this concretely: the experimentally measured rate law, Rate = k[H2O2][I-], matches neither reactant's own stoichiometric coefficient directly, but is explained by two bimolecular elementary steps, H2O2 + I- to H2O + IO- (slow, forming the intermediate IO-) followed by H2O2 + IO- to H2O + I- + O2 (fast). Since the first step is the slow, rate-determining one, and it alone involves both H2O2 and I-, it alone explains the overall reaction's observed first-order dependence on each.